The rift in the Republican Party grew deeper on Sunday and threatened to upset the July convention as Donald J. Trump refused to rule out blocking Paul D. Ryan, the speaker of the House, from serving as the convention’s chairman.

Mr. Trump’s warning was his latest affront to Republicans who have urged him to adopt a more cooperative and unifying tone. And it amounted to an extraordinary escalation in tensions between the party’s presumptive nominee and its highest-ranking officeholder.

In a series of television interviews that aired Sunday, Mr. Trump demonstrated little interest in making peace with party leaders like Mr. Ryan who have called on him to more convincingly lay out his commitment to the issues and ideas that have animated the conservative movement for the last generation.

“I’m going to do what I have to do — I have millions of people that voted for me,” Mr. Trump said on ABC’s “This Week.” “So I have to stay true to my principles also. And I’m a conservative, but don’t forget, this is called the Republican Party. It’s not called the Conservative Party.”

If anything, Mr. Trump’s candidacy has thrived because of his resistance to party politics as usual, not in spite of it. He has broken with Republican leadership in Congress on trade, military intervention and immigration policy. And he appears as determined as ever not to fall in line now that he has effectively secured the nomination.

Mr. Trump’s differences with those in the party who think they have earned more of a right to set its political and ideological course have led to a rupture at the time when Republicans would ordinarily be trying to put the messy personal clashes of the primary contests behind them.

These divisions have played out most openly and vividly around the planning of the Republican National Convention. It is a telling reflection of the state of Republican politics: an escalating spat over going to a party for a party that is coming undone.

Four of the last five Republican presidential nominees — George Bush, George W. Bush, John McCain and Mitt Romney — have said they will skip the convention in Cleveland, where Mr. Trump is expected to be formally nominated.

Mr. Ryan, who serves as the convention’s ceremonial chairman, has made the provocative declaration that he is not ready to support his party’s likely nominee, a rebuke that drew Mr. Trump’s threat, in an interview with NBC News, to keep him from assuming that role.

The large corporations that usually fund both parties’ conventions have grown wary of becoming involved. They are holding back on sponsorships, leaving Cleveland about $7 million short of its $64 million fund-raising goal just 10 weeks before the festivities begin.

“Conventions have always been platforms for different views inside the party, with the understanding that primaries are about our differences and the general election is supposed to be about coming together,” said Kevin Madden, a Republican consultant who has worked on several presidential campaigns, most recently in 2012 for Mr. Romney.

“But the big, open rift that nobody can deny is that there’s a lot we still don’t agree on,” he said.

With Mr. Trump’s two remaining rivals, Senator Ted Cruz of Texas and Gov. John Kasich of Ohio, now out of the race, Republicans have defused their biggest possible crisis ahead of the convention: having a bitter fight for the nomination play out in front of tens of millions on television. But they seemed to be barreling toward another.

Questions over Mr. Trump’s conservative credentials refuse to die, causing some Republicans to make demands of him that are unheard-of for the party’s standard-bearer.

Conservative activists have called on Mr. Trump to identify before he arrives in Cleveland people he would appoint as cabinet members, Supreme Court justices or even vice president, gestures they say would calm fears over the sincerity of his conservatism.

Tony Perkins, a convention delegate from Louisiana who is the president of the conservative Family Research Council and had supported Mr. Cruz before he dropped out of the race last week, said Mr. Trump could not afford to antagonize any more voters.

“The margins he has to work with in terms of electoral success are very small,” he said. Unlike other Republican nominees who have been greeted skeptically by social conservatives, Mr. Trump faces deep and unrelenting hostility, Mr. Perkins added. “Now, not only do you have indifference, you have outright resistance to his candidacy,” he said.

Having a nominee who engenders such mistrust poses complications for other aspects of the convention. As much as the party gatherings are meant to convey cohesiveness and cooperation, they have also become platforms to highlight diversity and inclusiveness, virtues Mr. Trump has not shown an inclination to promote.

Republicans have filled their speaking slots at recent conventions with women, African-Americans and Hispanics in an effort to overcome an image as the party of old white men. But the list of speakers from the 2012 convention reads like a list of Mr. Trump’s enemies. Many have denounced him, including Gov. Nikki R. Haley of South Carolina, Gov. Brian Sandoval of Nevada, Senator Kelly Ayotte of New Hampshire and Representative Barbara Comstock of Virginia.

And many of them are likely to skip the convention this summer. A spokesman for Mr. Sandoval said he did not plan to go to Cleveland. Ms. Comstock and Ms. Ayotte will stay home and meet with constituents instead. Ms. Haley’s office has said she has not made up her mind.

Convention organizers in the past have also invited leaders of different faiths, including Islam, to lead the body in prayer.

The stage usually serves as a forum for the party to help elevate its next generation of leaders. But there, too, the list from 2012 is a who’s who of officials whom Mr. Trump has alienated: Senator Marco Rubio of Florida, Gov. Scott Walker of Wisconsin and, of course, Mr. Ryan and Mr. Cruz.

Mr. Trump and Mr. Ryan are set to meet privately in Washington on Thursday as part of an effort to bridge the gap between the party establishment and Mr. Trump.

How to handle Mr. Cruz — and the hundreds of delegates who will go to the convention pledged to vote for him — is another sensitive issue for Mr. Trump and his team as they plan for the convention.

But Mr. Cruz, who has given no indication about what he wants his delegates to do, would most likely benefit from building up good will in the party if he wants to run for president again, as many expect him to do.

“It is in Senator Cruz’s interests to eventually be visibly supportive of Donald Trump as the Republican nominee,” said Paul Manafort, a senior adviser for the Trump campaign who was brought on to help improve outreach to fellow Republicans.

The Trump campaign must also begin raising money to plug any holes in the convention budget, which will exceed $100 million when costs like security are factored in. But some previous corporate sponsors, like Coca-Cola and Walmart, have been reassessing their commitments. Joe Roman, the vice chairman of the Cleveland host committee, said the large national corporations that were needed to close the $7 million gap the city is facing were taking some time to line up.

“The last five, six, seven million of anything is always the slowest,” he said.

There is also the question of safety. While the anger from Trump supporters who were fearful he would be denied the nomination appears to have diminished, masses of anti-Trump demonstrators are still expected to descend on Cleveland, a city that has been a caldron of racial tensions between the police and residents.

So even if many are staying away from the Republican National Convention, there is at least one group eager to go.